A072226 -id:A072226 - cp's OEIS frontend

A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 1, 2, 1, 6, 1, 1, 2, 5, 1, 1, 1, 1, 1, 8, 34, 8, 1, 2, 1, 10, 1, 2, 350, 2, 1, 111, 4, 1, 3, 16, 4, 15, 28, 3, 1, 206, 3, 10, 2, 1, 1, 2, 3, 1, 15, 637, 12, 1, 4, 22, 17, 104, 4, 2, 1012, 1, 1

Offset: 1

Eric Chen, Jan 18 2015

nonn

Conjecture: a(n) > 0 for all n.
a(n) is currently unknown for n = 121, 124, 143, 162, 171, 172, 185, 188, 197, 215, ..., for which we have n * a(n) > 130000.
a(121) = (A117545(2048))/11 and they are both currently unknown.
A117545(2^n) = a(A064549(n)).
a(130) = 917, a(144) = 820, a(164) = 720, a(201) = 606. - Max Alekseyev, Dec 04 2024

Eric Chen, Table of n, a(n) for n = 1..120
Eric Chen, Table of n, a(n) for n = 1..300 status

Table[k=1; While[!PrimeQ[Cyclotomic[n*k, 2]], k++]; k, {n, 43}]

a(n) = {k = 1; while (!isprime(polcyclo(k*n, 2)), k++); k;} \\ Michel Marcus, Jan 18 2015

A317299 Semiprimes in A072226.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 22, 26, 33, 34, 38, 46, 49, 62, 65, 69, 77, 85, 86, 93, 122, 129, 133, 145, 158, 202, 254, 334, 382, 398, 447, 471, 579, 626, 694, 745, 1402, 1727, 1781, 2353, 3415, 3418, 3481, 3817, 5053, 5234, 5403, 7078, 7617, 8033, 10967, 11581

Offset: 1

Jianing Song, Jan 22 2019

nonn

Semiprimes k such that A019320(k) is prime.

Numbers of the form p^2 where (2^(p^2) - 1)/(2^p - 1) is prime, or numbers of the form p*q where (2^(p*q) - 1)/((2^p - 1)*(2^q - 1)) is prime. Here p and q are necessarily primes.

			15 is a semiprime and Phi_15(2) = (2^15 - 1)/((2^3 - 1)*(2^5 - 1)) = 151 is prime, so 15 is a term. Here Phi_n is the n-th cyclotomic polynomial.
49 is a semiprime and Phi_49(2) = (2^49 - 1)/(2^7 - 1) = 4432676798593 is prime, so 49 is a term.

Jianing Song and Max Alekseyev, Table of n, a(n) for n = 1..79 (using data from A072226)

Cf. A019320, A072226.

for(k=1, 1000, if(isprime(polcyclo(k, 2))&&bigomega(k)==2,print1(k, ", ")))

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

A161508 Numbers k such that 2^k-1 has only one primitive prime factor.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208

Offset: 1

T. D. Noe, Jun 17 2009

nonn

Also, numbers k such that A086251(k) = 1.
Also, numbers k such that A064078(k) is a prime power.
The corresponding primitive primes are listed in A161509.
The binary expansion of 1/p has period k and this is the only prime with such a period. The binary analog of A007498.
This sequence has many terms in common with A072226. A072226 has the additional term 6; but it does not have terms 18, 20, 21, 54, 147, 342, 602, and 889 (less than 10000).
All known terms that are not in A072226 belong to A333973.

T. D. Noe, Table of n, a(n) for n=1..179
Wikipedia, Unique prime, section Binary unique primes.

Cf. A007498, A064078, A072226, A086251, A144755, A161509, A247071, A333973.

Select[Range[1000], PrimePowerQ[Cyclotomic[ #,2]/GCD[Cyclotomic[ #,2],# ]]&]

is_A161508(n) = my(t=polcyclo(n,2)); isprimepower(t/gcd(t,n)); \\ Charles R Greathouse IV, Nov 17 2014

A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 3, 6, 7, 9, 10, 12, 13, 14, 15, 21, 24, 26, 33, 36, 40, 46, 60, 63, 70, 71, 72, 86, 103, 108, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562, 665, 672, 680, 707, 718, 747, 760, 782, 828, 875, 892, 974, 984

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

Robert Price, Table of n, a(n) for n = 1..167
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019321, A072226, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 3]] &]

for( i=1,999, ispseudoprime( polcyclo(i,3)) && print1( i","))

A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408, 596, 612, 692, 732, 756, 800, 952, 996, 1004, 1228, 1268, 2240, 2532, 3060, 3796, 3824, 3944, 5096, 5540, 7476, 7700, 8544, 9800, 14628, 15828, 16908, 18480, 20260, 21924, 24656, 38456

Offset: 1

M. F. Hasler, Apr 03 2008

nonn

It appears that except for 1,2 and 6, all terms of this sequence are multiples of 4.
It also appears that all cyclotomic polynomials, Phi(k,x), where k is a multiple of 4 have no odd powers of x.  For example, Phi(20,x) = x^8 - x^6 + x^4 - x^2 + 1.  This implies that Phi(k,x) = Phi(k,-x), where k is a multiple of 4. - Robert Price, Apr 13 2012
Second comment is true; this follows from applying Theorem 1.1 in the Gallot paper with p = 2 and m even. - Charlie Neder, May 16 2019

Robert Price, Table of n, a(n) for n = 1..51
Yves Gallot, Cyclotomic polynomials and prime numbers
Index entries for cyclotomic polynomials, values at X

Cf. A019322, A072226, A138920-A138940.

Select[Range[1000], PrimeQ[Cyclotomic[#, 4]] &]

for( i=1,999, ispseudoprime( polcyclo(i,4)) && print1( i","))

a(29)-a(51) from Robert Price, Apr 12 2012

A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Phi_n(x) is the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nL(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
Let L(n) = the Aurifeuillian L-part of 2^n+1, L(n) = 2^(n/2) - 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let L*(n) = GCD(L(n), J*(n)).
This sequence lists all n such that L*(n) is prime.

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 and is not prime.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A297412 Numbers k such that A019320(k) is in A217468.

Original entry on oeis.org

43, 114, 163, 258, 326, 379, 487, 758, 762, 883, 974, 978, 1459, 1766, 2274, 2647, 2918, 2922, 3079, 3943, 5294, 5298, 5419, 6158, 7886, 8754, 9199, 10838, 11827, 14407, 15882, 16759, 18398, 18474, 18523, 23654, 23658, 24967, 26407, 28814, 32514, 33518, 37046, 37339, 39367

Offset: 1

Max Alekseyev, Dec 29 2017

nonn

Set difference of A297413 and A072226.

is_A297412(n) = my(m=polcyclo(n, 2)); Mod(2, m*(m-1))^m==2 && !ispseudoprime(m);

A250201 Least b such that Phi_n(b, b-1) is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 3, 2, 3, 4, 2, 6, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 4, 5, 40, 2, 3, 2, 7, 2, 5, 3, 3, 2, 13, 3, 2, 14, 4, 22, 3, 3, 13, 2, 34, 5, 3, 5, 2, 2, 34, 9, 2, 17, 7, 3, 2, 3, 18, 9, 47, 4, 20, 3, 2, 2, 8, 2, 4, 17, 6, 14, 2, 2, 61, 18, 2, 2

Offset: 2

Eric Chen, Mar 09 2015

nonn

Phi_n(b, b-1) = (b-1)^EulerPhi(n) * Phi_n(b/(b-1)).
This sequence is not defined at n = 1 since Phi_1(b, b-1) = 1 for all b, and 1 is not prime. Conjecture: a(n) is defined for all n>1.
If b = 1, then Phi_n(b, b-1) = 1 for all n, and 1 is not prime, so all a(n) > 1.
a(n) = 2 if and only if n is in A072226.
n        Phi_n(a, b)
1        a-b
2        a+b
3        a^2+ab+b^2
4        a^2+b^2
5        a^4+a^3*b+a^2*b^2+a*b^3+b^4
6        a^2-ab+b^2
...      ...
n        b^EulerPhi(n)*Phi_n(a/b)

			a(11) = 6 because Phi_11(b, b-1) is composite for b = 2, 3, 4, 5 and prime for b = 6.
a(37) = 40 because Phi_37(b, b-1) is composite for b = 2, 3, 4, ..., 39 and prime for b = 40.

Eric Chen, Table of n, a(n) for n = 2..490

Cf. A103794, A253633, A085398, A058013.

Table[k = 2; While[!PrimeQ[(k-1)^EulerPhi(n)*Cyclotomic[n, k/(k-1)]], k++]; k, {n, 2, 300}]

a(n) = for(k = 2, 2^16, if(ispseudoprime((k-1)^eulerphi(n) * polcyclo(n, k/(k-1))), return(k)))

cp's OEIS Frontend

A250209 a(n) = least k such that k * n is in A072226, or 0 if no such k exists.

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A317299 Semiprimes in A072226.

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A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

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A161508 Numbers k such that 2^k-1 has only one primitive prime factor.

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A138933 Indices k such that A019321(k)=Phi[k](3) is prime, where Phi is a cyclotomic polynomial.

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A138934 Indices k such that A019322(k) = Phi[k](4) is prime, where Phi is a cyclotomic polynomial.

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A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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